Graphics Reference
In-Depth Information
y
Consider Figure 26.14, which shows light arriving at the interface between two
media, the upper (
y
0) having
index
n
2
. For now, we'll assume that the media are insulators rather than con-
ductors. The light's direction of propagation lies in the
xy
-plane, the plane of the
diagram. The arriving light makes angle
>
0) having refractive index
n
1
and the lower (
y
<
i
r
Index
n
1
θ
i
(“i” is for “incoming”) with the
y
-axis;
the reflected light makes angle
θ
t
with the negative
y
-axis. Since the electric field associated to the incoming light
must be perpendicular to the direction of propagation, we'll consider two special
cases. In the first, the electric field, at each point of the incoming ray, points along
the
z
-direction (i.e.,
parallel
to the interface between the media, pointing either
into or out of the page). A light source with this property is said to have “parallel”
polarization with respect to the surface, or be
p-polarized.
θ
r
=
θ
i
, and the transmitted light makes angle
Index
n
2
t
Figure 26.14: A light ray reflects
and transmits through an inter-
face between media.
When such a wave reaches the surface the electric field interacts with the
electrons near the interface, moving them back and forth in the
z
-direction; these
motions in turn generate a new electromagnetic field that's a sum of two parallel-
polarized waves, the first corresponding to the transmitted light and the second to
the reflected light. The transmitted light travels in a direction described by Snell's
law, and the reflected light travels according to the familiar “angle of incidence
equals angle of reflection” rule:
θ
r
=
θ
i
.The
fraction R
p
of light
reflected
depends
θ
i
according to the rule
on the angle
r
p
=
n
2
cos
θ
i
−
n
1
cos
θ
t
(26.10)
n
2
cos
θ
i
+
n
1
cos
θ
t
R
p
=
r
p
.
(26.11)
the fraction transmitted
T
p
is just 1
R
p
. (These fractions denote the fraction of
the incoming
power
that leaves in each direction. The
amplitude
of the reflected
wave is just
r
p
times the amplitude of the arriving wave.) These formulas can be
derived, like Snell's law, by insisting on continuity at the interface [Cra68].
The
phase
of the reflected light may match that of the arriving light, lag behind
it, or lead it, or be 180° out of phase with it.
The other special case is when the electric field is perpendicular to the
z
-axis,
that is, it lies entirely in the
xy
-plane, perpendicular to the direction of propagation.
Such a wave is said to be
s-polarized.
In this case, the reflection coefficient
R
s
is
given by
−
r
s
=
n
1
cos
θ
i
−
n
2
cos
θ
t
(26.12)
n
1
cos
θ
i
+
n
2
cos
θ
t
R
s
=
r
s
(26.13)
R
s
. These rules for the
reflection and transmission coefficients for s- and p-polarized waves are called
the
Fresnel equations,
after Augustin-Jean Fresnel (1788-1827).
Because every wave can be written as a sum of an s-polarized and a p-polarized
wave, these two special cases in fact tell the whole story. For instance, incoming
light that is linearly polarized as the sum of a wave that is equal parts s-polarized
and p-polarized will reflect and again be linearly polarized. But the ratio of
s
-to
Once again, the transmission coefficient
T
s
is 1
−
